1. Prime Factorization and Number Theory 1.1) Express 660 as a product of its prime factors. - Start dividing 660 by the smallest prime numbers: $$660 \div 2 = 330$$ $$330 \div 2 = 165$$ $$165 \div 3 = 55$$ $$55 \div 5 = 11$$ $$11 \div 11 = 1$$ - So, the prime factors are: $$660 = 2^2 \times 3 \times 5 \times 11$$ 1.2) Find HCF and LCM of 9860 and 2958 given their prime factorizations: - 9860 = $$2^2 \times 5 \times 17 \times 29$$ - 2958 = $$2 \times 3 \times 17 \times 29$$ - HCF is product of common prime factors with lowest powers: $$HCF = 2^1 \times 17 \times 29 = 986$$ - LCM is product of all prime factors with highest powers: $$LCM = 2^2 \times 3 \times 5 \times 17 \times 29 = 29580$$ 2. Proportions and Ratios 1.3) Volume and Pressure Relationship (a) Variables are inversely proportional because as volume increases, pressure decreases. (b) Use inverse proportionality formula: $$P \times V = k$$ - For volume 0.4 and pressure 300: $$k = 300 \times 0.4 = 120$$ - Calculate missing pressure a at volume 0.8: $$a = \frac{k}{0.8} = \frac{120}{0.8} = 150$$ - Calculate missing pressure b at volume 6 (already given as 40, check consistency): $$b = \frac{k}{6} = 20$$ (Given 40, so b is 20 if consistent) 1.4) Solve proportion: $$5 : c = d : 20 = 60 : 88$$ - From $$5 : c = 60 : 88$$, cross multiply: $$5 \times 88 = 60 \times c \Rightarrow c = \frac{440}{60} = \frac{22}{3}$$ - From $$d : 20 = 60 : 88$$, cross multiply: $$d \times 88 = 60 \times 20 \Rightarrow d = \frac{1200}{88} = \frac{300}{22}$$ 1.5) Mass of green blocks in 2kg bag with ratio 5:3:2 - Total parts = 5 + 3 + 2 = 10 - Mass of green blocks = $$\frac{3}{10} \times 2 = 0.6$$ kg 3. Speed, Commission, VAT, Discounts, Interest 1.6) Average speed = $$\frac{distance}{time}$$ - Distance = 12000 m = 12 km - Time = 270 minutes = 4.5 hours - Speed = $$\frac{12}{4.5} = 2.67$$ km/h 1.7) Commission = 5% of 12,500,000 - Commission = $$0.05 \times 12500000 = 625000$$ 1.8) Price excluding 15% VAT - Let price excluding VAT = x - Price including VAT = x + 0.15x = 1.15x = 5.70 - So, $$x = \frac{5.70}{1.15} = 4.96$$ 1.9) Find base amount where 46 is 115% - Let base = x - $$1.15x = 46 \Rightarrow x = \frac{46}{1.15} = 40$$ 1.10) Price after 30% discount on 18 - Discount = 0.30 \times 18 = 5.4 - New price = 18 - 5.4 = 12.6 1.11) Simple interest on 15000 at 6% for 18 months - Time in years = $$\frac{18}{12} = 1.5$$ - Interest = $$P \times R \times T = 15000 \times 0.06 \times 1.5 = 1350$$ 1.12) Compound interest for 15000 at 12% for 3 years - Formula: $$A = P(1 + i)^n$$ - $$A = 15000(1 + 0.12)^3 = 15000 \times 1.404928 = 21073.92$$ 4. Simplification Without Calculator 2.1) $$-2 + 3 - 6 - (-4) + (-1) = -2 + 3 - 6 + 4 - 1 = -2$$ 2.2) $$5 \times 1 - 1 + 3 \times (-9) = 5 - 1 - 27 = -23$$ 2.3) $$11 \times 3 - 15 + 5 = 33 - 15 + 5 = 23$$ 2.4) $$-2(\sqrt{16} + \sqrt{8} - 3^2) = -2(4 + 2\sqrt{2} - 9) = -2(-5 + 2\sqrt{2}) = 10 - 4\sqrt{2}$$ 2.5) $$16 \div (2 - 4)^3 = 16 \div (-2)^3 = 16 \div (-8) = -2$$ 5. Exponent Laws 3.1) $$\frac{32a^7 \times 64a^5}{512a^6} = \frac{(2^5)a^7 \times (2^6)a^5}{2^9 a^6} = \frac{2^{11} a^{12}}{2^9 a^6} = 2^{2} a^{6} = 4a^6$$ 3.2) $$10 \times 100 \times 1000 \times (-1)^3 = 10 \times 10^2 \times 10^3 \times (-1) = 10^{6} \times (-1) = -1000000$$ 3.3) $$(b^3 \times c^4 \times c^{-6})^3 = (b^3 \times c^{-2})^3 = b^{9} c^{-6}$$ 3.4) $$16^{2/3} = (2^4)^{2/3} = 2^{8/3}$$ and $$15^0 = 1$$ 3.5) $$\left(\frac{3x^{-5}}{9x^7}\right)^{-2} = \left(\frac{1}{3} x^{-12}\right)^{-2} = 3^{2} x^{24} = 9x^{24}$$ 6. Linear Pattern 4.1) Pattern: 7, 25, 43, ... - Difference between terms: 18 4.1.1) Next term = 43 + 18 = 61 4.1.2) Formula for nth term: - $$T_n = a + (n-1)d = 7 + (n-1)18 = 18n - 11$$ 4.1.3) 951st term: - $$T_{951} = 18(951) - 11 = 17118 - 11 = 17107$$ 4.1.4) Check if 22255 is a term: - Solve $$18n - 11 = 22255$$ - $$18n = 22266 \Rightarrow n = \frac{22266}{18} = 1237$$ - Since n is an integer, 22255 is a term in the pattern.